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Marie-Edith Bissey and Guido Ortona (2002)

The Integration of Defectors in a Cooperative Setting

Journal of Artificial Societies and Social Simulation vol. 5, no. 2
<http://jasss.soc.surrey.ac.uk/5/2/2.html>

To cite articles published in the Journal of Artificial Societies and Social Simulation, please reference the above information and include paragraph numbers if necessary

Received: 10-Dec-2001      Accepted: 17-Mar-2002      Published: 31-Mar-2002


* Abstract

This paper describes a study of the robustness of cooperative conventions. We observe the effect of the invasion of non-cooperating subjects into a community adopting a cooperative convention. The convention is described by an indefinitely repeated prisoner-dilemma game. We check the effects on the robustness of the cooperating convention of two characteristics of the game, namely the size of the prisonner-dilemma groups and the "intelligence" of the players. The relevance for real-world problems is considered. We find that the "intelligence" of the players plays a crucial role in the way players learn to cooperate. The simulation program is written in SWARM (Java version).

Keywords:
Conventions; Cooperation; Prisoner's Dilemma; Social Simulation; SWARM

* Introduction

1.1
This paper describes a simulation frame to study the robustness of cooperative conventions. In this section we will discuss three introductory points: the class of real-world problems this approach may be useful to deal with; (some of) the limits of this approach; and the specific topics that our program allows us to investigate.

1.2
We assume that cooperative conventions play a very basic role in an organized society[1]. When two (or more) societies meet, the conventions of both are challenged. When they finally merge, a new society will result, characterized by a new set of conventions. However, this new set may be of a different nature than the previous one. With reference to the integration of people forming the societies, the range of possible conventions, both logically and historically, goes from the "melting pot" of Roman, Austrian and (possibly) American empires to the full apartheid of South Africa's Blacks, India's Pariahs and Europe's Gypsies. The final result will depend upon a lot of factors; the nature of the cooperating conventions (the concept will be discussed below) is likely to be a very relevant one.

1.3
To our knowledge, up to now there are no experimental or simulative studies on this subject. Probably, Kirchkamp (2000) is the closest one. There is an interesting and large literature on a related topic, that of the stability of a cooperative equilibrium in a game that may be characterized as a convention. The possibility of a stable cooperative equilibrium is an acquired result in economics (see for instance Sugden, 1986 and 1989; Axelrod, 1981 and 1986; Witt 1986 and 1994). Recently, some authors have started to try to point out the characteristics that may actually drive the players towards a stable cooperative equilibrium. A very brief discussion of some of this literature may be useful, mostly because it will help to clarify the limits of our approach.

1.4
Vogt (2000) takes an analytical approach. He assumes a population where both cooperators and non cooperators are present, and where a mutation allows some subjects to record the nature of the partner. These subjects cooperate with cooperators, and defect with defectors. It results that in equilibrium it is possible for cooperators and defectors to coexist.

1.5
Eshel et al.(2000) and Cooper and Wallace (2000) take a simulative approach. The first shows that cooperation may result as a stable strategy (see below) if players interact with neighbours and imitate successfull players. The second shows that cooperation may result if players can choose their partners.

1.6
These three papers - and others - make ad hoc assumptions, to show that cooperation may arise spontaneously. Their point is that simple, plausible, yet unexplained behavioral assumptions may be sufficient to get out of the Hobbesian state of nature. Our paper is analogous. In other words, we will not attempt to build a "golem" artificial society[2] to look at what may go on in it. Instead, we will investigate what characteristics of cooperation make it robust with reference to some possible setting of the society. As previous studies showed, we expect our results to be strongly influenced by the simplistic nature of our fictitious world, and therefore we will be cautious when transferring them to the real world. In a sense, our results will be in the nature of conjectures. However, we hope that they may be of interest, for two reasons. The first is that, with reference to the important problem of the conflict of conventions, we lack even conjectures. The second is that in an era of globalization the issue is of a great practical relevance, so conjectures may be useful to calibrate on-field inquiries. A (preliminary) list of the questions we think can be investigated within our approach illustrates the point.

  1. How is the robustness of a cooperative convention affected by the number of cooperators required to implement it?
  2. What kind of cooperative convention is more robust?
  3. What is the relative weight of gain from cooperation and of learning in defending cooperative conventions?

1.7
The simulations presented in this paper illustrate mainly point a). However, the program already allows to look into b) and c). Section 2 contains a very brief presentation of the theory of conventions and describes the model. Section 3 shows results of our simulations. Conclusions follow. The simulations are written in SWARM. A technical appendix illustrates briefly the simulation program.

* Definitions and Method

Conventions

2.1
The concept of "convention" enjoys a rigorous definition in the social sciences. According to Sugden (1986, p.32), a convention is any stable equilibrium in a game that has two or more stable equilibria. A formal statement follows. This definition is remarkably powerful, for two reasons. First, it corresponds quite well to the common sense meaning (see below). Second, it is formally identical to the definition of an Evolutionary Stable Strategy (ESS) used in evolutionary biology (see for instance Maynard Smith, 1982).This throws several useful bridges between economics and biology, for instance the comparison of learning and adaptation and the conditions for multiple equilibria. Formally, the definition of a convention is the following, according to Sugden (1986):

Given an indefinitely repeated non cooperative game, a strategy I is a convention if all the players adopt I and for every player
(a) E(I,I) = E(J,I)
(b) if E(I,I) = E(J,I), E(I,J) > E(J,J).
for every strategy J≠I, where E(a,b) is the expected payoff of a player adopting strategy a against a player adopting strategy b.
(c) There is more than one strategy satisfying conditions (a) and (b).

2.2
Conditions (a) and (b) define an ESS; condition (c) is peculiar to the social sciences context. The definition states that if strategy I satisfies the first two conditions, it is stable, i.e. cannot be invaded, in the sense that no player may do better adopting another strategy. Hence it defines a behavioural rule that is self-sustained, with no need of an external enforcement device. This is the first half of the common sense definition of a convention. The second half stems from condition (c): a convention is such because there are many possible rules, and to choose one is properly a matter of convention. A famous classroom instance is to "drive on the right side of the road": the rule to "drive on the left side of the road" works as well.

2.3
Usually, conventions are studied in the context of a prisoner's dilemma game (PD), and here we will do the same. The reason is twofold. On one hand, the game is very powerful and evocative in describing social dilemmas. On the other hand, it has been shown that the game allows for cooperative conventions to take place: in an indefinitely repeated PD there is at least a family of strategies that prescribes cooperation, such that, once adopted, they cannot be invaded. This family is the famous set of tit-for-tat strategies, albeit without its simplest member. As there are many such strategies, and in addition "never cooperate" also cannot be invaded, all of them qualify as conventions. Note, in addition, that there is no limit to the complexity the definition allows for. For example, strategy I may prescribe "do X if you are in condition a, but Y if you are in condition b". Obviously, not only the number of simple strategies may be large (or infinite), but the number of players too. This is why it is usually believed that complicated cooperative patterns may be reduced to the basic frame outlined above, or, from another and more suggestive point of view, that whenever there is a cooperative behavior not enforced by an exogenous constraint, there is a cooperative convention at work.

2.4
The last sentence looks plausible, yet it raises some relevant questions. What do we intend by "exogenous enforcing constraint"? In the game-theoretic literature it is usually assumed that a convention is actually sustained by a punishment; the punishment, however, is not applied by a specific subject who is not a player, but spontaneously by the other players themselves. This result is theoretically sound. For instance, Axelrod (1986) shows that once a metanorm calling for retaliation against defectors has evolved, it may constitute in turn a stable equilibrium. Analogously, if cooperation is supported by a tit-for-tat strategy, a defector must suffer some rounds of cooperation against non-cooperators, and cannot do better refusing this punishment (i.e, non-cooperating in turn; see for instance Sugden 1986). In our model we will adopt an Axelrod-type punishment rule. However, two important problems remain unresolved. First, we cannot assume that the rules of punishment adopted are exogenous to the nature of the game. Second, the difference between formal and spontaneous enforcement devices must be characterized with reference to real world situations[3].

2.5
We will not go deeper into this. Both problems relate to the topic of spontaneous order in general, and are not specific to our theme. Our paper simulates a case of conflicting conventions according to the game-theoretic definition. The validity of its results for real-world cases is subject to the same caveats that hold in general for results concerning the game-theoretical approach to spontaneous order. From this point of view too the extension of our results to the real world may be only conjectural, as we suggested, for other reasons in the previous section.

The Model

2.6
Our model consists of a "world" (defined by a 2-dimensional space), populated by a fixed number of subjects. We assume that the initial population of this world (Natives [N]) is made of subjects that move randomly in the space, and meet with their neighbours (which change with the moves)[4]. When they meet, the subjects play a Prisonner's Dilemma (PD) game. We assume that this native population has evolved to play "cooperation" systematically. This evolution stems from the fact that subjects get punished if they do not cooperate with a cooperator, hence it pays more to cooperate. As we saw, this is how cooperative conventions are normally supposed to work. A player chooses the strategy providing the highest expected payoff; in this setting, it is always better to cooperate[5].

2.7
At the beginning of the simulation, the world has been invaded by new subjects (Immigrants [I]). We assume that Immigrants have a similar history as Natives, except for the fact that they never learned to cooperate in the PD. Hence, they arrive in our world being used to a non-cooperative convention. They also choose what to do according to the highest expected payoff[6].

2.8
As before, Immigrants and Natives will meet and play the PD game. Now it may become preferable for subjects N not to cooperate (if they expect the other player(s) to be I), and for subjects I to cooperate (if they expect the other player(s) to be N). Both kinds of players evaluate the expected payoffs of each meeting basing themselves on their memories of previous encounters, so the game evolves. After a while, either all players will have learnt to cooperate, or all will have learnt not to, or some will do and some not. What will happen depends upon the value of the parameters defining the relative number of immigrants, the payoffs, the punishment, the nature of the cooperation, the memory, the pattern of invasion and so on. The effect of these parameters is what we want to observe.

2.9
We assume a group invasion for two reasons. First, this feature is realistic with reference to the problems of the integration of ethnic minorities. Second, this way we may suppose that payoffs of the model do not change. If the invasion is made by individuals, as for instance in Uno and Namatame (1999), a convention may be changed only if payoffs change. However, if payoffs change anything may happen, and obtaining general results may become too difficult.

The Simulation

2.10
The program is described at some length in the appendix. To anticipate some details:

(a) The simulation is based on a spatial prisoner's dilemma where players meet with their neighbours, play the PD game and move to another part of the space at the end of each round[7]. Players are located in the space according to three specifications:

  • Ghetto: Natives and Immigrants are initially segregated in given sectors of the space which they cannot leave. At the end of each round, they move randomly to another position, within their own space.
  • Random: natives and immigrants are mixed in random fashion in the space and can move anywhere.
  • Invasion: immigrants start in a Ghetto, then gradually "invade" the space at a rate we can parametrise. Coincidentally, natives can enter the initial ghetto at the same rate.

Table 1: The Space and how Players move in it

The space for the Invasion setting at the beginning: Immigrants are in red in the upper left hand side corner. Natives are in green. All players are disposed randomly on the space.
This disposition of players in the space is also the one used in the Ghetto setting.
The space for the Invasion setting after a few rounds of the simulation. Players have moved, and we can see that Immigrants have started to "invade" the Natives' space (and vice versa). In the picture, Immigrants who have learned to cooperate appear in blue. Isolated players who could not be part of any group and are sleeping for the current round are represented in white (Natives) or yellow (Immigrants).The space for the Invasion setting at the end of the simulation. Both Natives and Immigrants can move anywhere in the space (as in the Random setting). In this case, all immigrant players have learned to cooperate (no more players are red).

2.11
(b) We model four types of players, (Normal, Intelli, Clever and Nobel) according to their "intelligence", i.e. to their ability of taking into account the information conveyed by the story of the game.

  1. Normal players are unable to identify the type (Native or Immigrants) of the other members of the group. Hence they base their strategy decision on all their previous moves and on the payoffs that resulted.
  2. The three other types are able to identify the type of their opponents in the group, and will therefore take that information into account while making their strategy decision. Intelli players (like Normal players) remember their own moves and the resulting payoffs, as well as the composition of the group. For instance, if an Intelli Native player finds himself in a group with say 2 other Natives and 3 Immigrants, he will base his strategy decision on all his moves and payoffs obtained when he was in a group with 2 other Natives and 3 Immigrants. What happened when the group was different has no impact on his decision.
  3. Clever players will identify the number of Immigrants in the group, and remember how many times in the past those players defected. They do the same for the Natives in the group. This will enable them to compute a probability of encountering defection, which they use for making their strategy decision.
  4. Nobel players have a strategy similar to Clever players, with the addition that they remember the composition of the group. So if a Nobel player finds himself in a group with 2 other Natives and 3 Immigrants, he will recall all times in the past when the immigrants (or the natives) defected, when he was in a similar group situation, and will base his strategy decision on the resulting probability of encountering defection.

2.12
As the definitions above indicate, Intelli players are an extension of Normal players (they have the same way of making their decision, but a more complex memory). Similarly, Nobel players are an extension of Clever players.

2.13
(b) We observe the resistance of the initial cooperating convention to the arrival of immigrants using various sizes of PD playing groups.

(c) We also vary the benefits of cooperating in the society, as follows.
  1. First setting: cooperating yields benefits (strictly positive payoff) even if some members of the PD group defect, and the size of the maximum payoff obtained when cooperating increases with the size of the group;
  2. Second setting: cooperating is ineffective (payoff becomes 0) unless all members of a group cooperate, and the size of the maximum payoff obtained when cooperating increases with the size of the group;
  3. Third setting: cooperating yields benefits as in setting 1, but the size of the maximum payoff obtained when cooperating is fixed .

2.14
Table 2 shows the payoff values of the settings we have considered. In this table, n is the size of the group of players. The first column shows the number of defectors in the group (from 0, "everybody cooperates" to n, "everybody defects"). We see therefore that the benefits of cooperating (and defecting) increase with the size of the group for settings 1 and 2, while it is fixed in setting 3. Being the sole defector in a group is always better than cooperating (in the absence of punishment). In settings 1 and 3, cooperators earn a positive payoff even if there are defectors in the group, while in the second setting, cooperation yields a payment only if all members of the group cooperate. In addition, all three payoff settings yield the same earnings in groups of size 2.


Table 2: The three payoff settings

Number of
Defectors
First setting payoffsSecond setting payoffsThird setting payoffs
DefectorsCooperatorsDefectorsCooperatorsDefectorsCooperators
0--n--n--2
1n+1n-2n+1032((n-2)/n-1))
in-(i-1)n-(i+1)n-(i-1)032((n-(i+1))/(n-1)
n1--1--1--

2.15
An important characteristic of our model is punishment which happens to any player who defects when some player(s) in the group cooperate(s). In this case, the defecting player earns what he would have earned had he cooperated, minus one (which, for payoff setting 2, means a loss). Such a punishment will tend to favour cooperation over defection. Table 3 shows the payoffs of cooperators and defectors with the second payoff setting in a group of size 5. When punishment is included in the model, we see that defectors earn a negative payoff (-1) if some players in the group have cooperated. When all players in the group defect, there is no punishment, so they earn the PD's payoff (1).


Table 3: Example of punishment for Payoff Setting 2, in a group of size 5

Number of defectors
(size of group: 5)
Payoff defectors
(no punishment)
Payoff cooperatorsPayoff defectors
(with punishment)
0--5--
160-1
240-1
330-1
420-1
5101

We model how participants are "attached to their traditions". This feature is represented by a number of periods in which each type of participant is supposed to have encountered only his `traditional' behaviour situation (all cooperate for natives, all defect for immigrants), before the game starts. This ensures that at least in the first round of the game, players will play according to their tradition and will only learn that some players are different through meeting them.

* Results

Results with Payoff Setting 2

3.1
We think that Setting 2 is more realistic, and hence more interesting: cooperation is ineffective unless all the members of the group cooperate -as in a typical productive unit of an economy based on the division of the labour. The payoff for cooperators also increases with the group size. So we will first consider the results for Setting 2. We assume 5000 players, 5 per cent of whom are immigrants. As N(ative) subjects always cooperate, I(mmigrants) will sooner or later learn to cooperate too[8]. What we observe is how long this time span is, with reference to three features/parameters: memory, group size, and spatial diffusion. We read them as metaphors of the real world as follows:

  • Memory: a longer memory means a stronger attachment to traditions. We consider the values of 1 (the player "remembers" only 1 period before the simulation started) and 10 (the player "remembers" 10 periods before the simulation started). Recall that Memory is used to weigh the current experience of the players in the simulation. Hence, the larger the value of Memory, the smaller the relative weight of the results obtained in the simulation.
  • Group size: this is a "technology" feature. A larger group size implies that more cooperators are needed to obtain the gains from cooperation. Hence there is more division of labour, and the society is more complex and more integrated.

3.2
As for the spatial pattern, we consider only Random and Invasion3 (that is, Invasion with a speed of progress of 3 cells per round)[9]. We consider all four types of "intelligence" (Normal, Intelli, Clever and Nobel). Results appear in Table 4. They are average results of 6 simulations each with a different random seed[10]. These simulations present the results of the random and invasion (with invasion speed of 3) models.


Table 4: Number of rounds for Immigrants to learn to cooperate (payoff setting 2)

PAYOFF SETTING 2Group Size
MemoryPopulationPosition2345678910
NORMAL
15000random222232333
15000invasion3666667777
105000random655444433
105000invasion3111010111111111111
INTELLI
15000random222333445
15000invasion367891011121213
105000random65655671622
105000invasion3101215151819222425
CLEVER
15000random222222222
15000invasion3655444444
105000random222222222
105000invasion3655444444
NOBEL
15000random222333334
15000invasion366778891010
105000random222333334
105000invasion366778891010

3.3
This format of the data emphasises the time required for cooperative behavior to become universal. We note that:

  • Memory: a longer memory/attachment to traditions makes convergence to universal cooperation take longer for Normal players. This is not the case for more intelligent players.
  • Spatial pattern: if I subjects come out dispersed (random instead of invasion), integration is easier.

3.4
These results were expected, and they only confirm that the model is not too much at odds with reality. The following one is more interesting.

  • Group size and intelligence of the players: the relationship of group size with the intelligence of the players for the speed of the convergence towards cooperation is not monotonic. For "Normal" players, increasing the group size does not appear to speed up cooperation in the population, and in some cases slows it down. For "Intelligent" players there is a direct relationship between the size of the group and the speed of cooperation. For clever players the pattern is similar to that of "Normal" players. Finally, for "Nobel" players, there appears to be a direct relationship between the size of the group and the speed of cooperation. In addition, small groups demand less time to produce integration if subjects are more intelligent. In our opinion, this reads as follows: if the society is simple (as depicted by the small size of the group required to implement cooperation), the more intelligent players get integrated more rapidly. But if the society is complex, the pattern is very mixed: subjects who are integrated most quickly are the "Clever"s, while "Normal"s and "Nobel"s behave approximately the same way up to the extreme of our range.

3.5
This result is both counterintuitive and in line with important pieces of empirical evidence. To cooperate gives in the long run a higher overall payoff: hence one would expect more intelligent players to move more quickly to cooperation. However, it is true that integration among ethnic communities is often more difficult when the economies are complex and the capability of the members of the minority group to understand the behavioral norms of the society is high. This feature has been related in previous research to the willingness of the ethnic leadership to exploit fully the human capital of the followers (see Ortona 2001, ch. 10; Breton and Breton 1995). Our model suggests (no more than that) that the features included in it (memory, intelligence, etc.) may be sufficient to produce this result. Obviously, this is a point that deserves further inquiry.

Comparing the Payoff Settings

3.6
Differences between the three payoff settings, or the lack of them, are apparent, and some of them are interesting. In two rows of Setting 1 for Normal players, the relationship between the speed of the convergence to cooperation and the group size seems to be the opposite of that of Setting 2. But in general, the differences between Setting 1 and Setting 2 are small. This is quite surprising, and we are not, at present able, to suggest an explanation. As for Setting 3, there are substantially no differences with Setting 2 for the more intelligent types of players, while for the Normal players, the relationship between time and group size is much stronger (with one exception). This means that it is more difficult to reach cooperation in Setting 3 when Immigrants are less able to process information. This difficulty is also increased when they are more linked to their traditions, as the difference is particularly high for a memory of 10. This result is not unexpected, since in Setting 2 the benefit from cooperation increases with the size of the group and in Setting 3 it does not. It suggests a conjecture quite important for real world policing: Immigrants will be integrated faster if they participate in activities where the individual contribution to the attainment of a group payoff is important. There are other differences, too, albeit minor, but we cannot think of intuitive explanations for them. This suggest that in the real world some unsuspected features of the relationship between groups may produce unexpected effects.


Table 5: Number of rounds for Immigrants to learn to cooperate (payoff setting 1)

PAYOFF SETTING 1Group Size
MemoryPopulationPosition2345678910
NORMAL
15000random222222222
15000invasion3655555444
105000random666766777
105000invasion3111111111111111111
INTELLI
15000random223333333
15000invasion366778891010
105000random678899111316
105000invasion3101415181921222324
CLEVER
15000random222222222
15000invasion3655444444
105000random222222222
105000invasion3655444444
NOBEL
15000random223333333
15000invasion366778891010
105000random223333333
105000invasion366778891010


Table 6: Number of rounds for Immigrants to learn to cooperate (payoff setting 3)

PAYOFF SETTING 3Group Size
MemoryPopulationPosition2345678910
NORMAL
15000random222222222
15000invasion3655555544
105000random6910101212121313
105000invasion3111214141516161516
INTELLI
15000random223333333
15000invasion366778891010
105000random61012131617192534
105000invasion3101621232629303535
CLEVER
15000random222222222
15000invasion3655444444
105000random222222222
105000invasion3655444444
NOBEL
15000random223333333
15000invasion366778891010
105000random223333333
105000invasion366778891010

* Conclusions

4.1
As all simulators know, the gap between a manageable but useless model and a nearly one-to-one mapping of reality, too specific to allow for general conclusions, is very narrow. To minimize the risk on either side, we preferred to try another way, i.e. to build a model that does not aim to be a picture of a real situation, but allows testing for some characteristics of abstract players with reference to a specific game-theoretical model. We hope to obtain conjectures about the relevance of some actual features in propagating, or hindering, the integration among ethnic groups. The first results look encouraging. In our opinion, the most suggestive is that the relationship between the broadness of the information the players may use (their "intelligence") and the spreading of cooperation is non-monotonic. We hope to obtain further indications through the introduction of new features, such as a more flexible immigration pattern, the possibility of distinguishing more fully the type of other group members, the presence of subjects with different types of intelligence, and the possibility of obtaining information through the observation of other players. All these features will be described by parameters; hence it will be possible to assess their effect through successive simulations.

* Technical Appendix

The Program[11]

5.1
The simulation program is written using the SWARM simulation language (Java version)<http://www.swarm.org/>. It operates by creating N subjects and setting up their intelligence, payment structure, way to move around the space, etc, according to the parameters explained below. The program then runs for either 100 rounds or until a convention is adopted by the whole population (whether it is a cooperating or a defecting one).

5.2
What happens in each round? Players are set in a 2-dimensional space, of which they occupy a cell. The space is chosen large enough so that "neighbourhoods" may exist, but also unoccupied cells (this is controlled by the densityIndex parameter, explained below). A neighbourhood can be defined as a set of players that "touch" each other by either a corner or a side of their cells. At the beginning of each round, all players are paired with their neighbours in order to form groups of a given size. So except for players situated on the borders of the space, each member of the population can have at most 8 neighbours and these in turn can have at most 8 neighbours, etc. If there are not enough neighbours (and neighbours of neighbours) to form a group of the required size, those players are told to "sleep" for the current round (they do not play the PD, and earn nothing). If there are enough players, they play the PD, and get paid according to the strategy they have chosen, the payoff setting and the behaviour of the other group members. Before starting another round, all players move randomly to a different position in their allocated space (which depends on the playerMover parameter).

5.3
The next section describes the parameters of the simulation, which can be modified by the user before each run. We will point out which of those are fixed (and to which values) during the simulations, and which vary.

List of Parameters

Global parameters of the simulation

sizeOfPopulation: total number of players in the game (an integer), fixed at 5000

sizePDGroups: size of the Prisonner's Dilemma groups (an integer, smaller than sizeOfPopulation), varied from 2 (the smallest possible) to 10

proportionImmigrants: total proportion of immigrants in the game (a number between 0 and 1), fixed at 5%

playerMover: how players are positionned in the space at the beginning of the game and how they move during the game:

* Ghetto: Immigrants and Natives are in their own part of the space (dimension proportional to their respective numbers). These spaces cannot be invaded. They move randomly within their own space. As a consequence, the only contact between them occurs at the frontier of the space.
* Random: Immigrants and Natives are dispersed randomly through the entire space, and move randomly. This option maximises the possibility of meetings between Immigrants and Natives.
* Invasion: This is a mixture of the previous two. Immigrants and Natives start in the Ghetto setting, and remain in it for a given number of rounds (timeInOwnSpace), after which they are allowed to "invade" each other's space (at a speed of invasionRate, which defines the number of "cells" by which the space of each player is increased). Players wait for some rounds (defined by timeBetweenInvasions) before increasing their space further. Eventually, they end up in the same situation as Random. We present the results for Random and Invasion with a speed of 3 cells.
densityIndex: This parameter is used to define the dimensions of the space on which players are positioned, which is proportional to sizeOfPopulation (the dimension is the square root of sizeOfPopulation multiplied by densityIndex), fixed at 2

endAfterRound: The game is based on an indefinitely repeated PD, and stops when an equilibrium is found (either all cooperate or all defect). However, it may happen that no equilibrium is ever found. This parameter allows stopping the simulation after a suitably large number of rounds in such cases, fixed at 100.

proportionIntelliImmigrants, proportionIntelliNatives, proportionClever-Immigrants, proportionCleverNatives, proportionNobelImmigrants, proportionNobelNatives: These parameters define the proportion of different types of players in the population. These proportions are used respectively with the total number of Natives and Immigrants in the population. If the sum for Intelli, Clever and Nobel immigrants does not sum to 1, the remaining players are Normals. The same occurs for Natives.
For the purposes of the simulation, all players in the population were of the same intelligence (i.e. all Normal, all Intelli, all Clever or all Nobel). We plan to investigate mixed populations in further research.

payoffSetting: This parameter identifies the type of payoffs players get from the game. It takes so far four possible values ("first", "second", "third" and "fourth"). Each setting defines a different way to reward (or not) cooperation in the face of defection. In the simulations, the "second" payoff setting was used, in which cooperation is only rewarded when all other players in the group are cooperating.

pastTimes: For all players, this parameter defines their "memory", which can be interpreted as their attachment to their traditions. Typically, it has a higher value for Immigrants than for Natives (making the former more reluctant to change from non-cooperation to cooperation). In the simulation, pastTimes for the Natives was always fixed to 1. For the Immigrants, it was alternately fixed to 1 and 10.

learningRate: In addition to the memory parameter, players can learn at different speeds. In the simulations, all players had a learning rate of 1.


* Notes

1This is an established result in modern economics (see for instance Sugden 1986); but it has been recognised only recently, despite the early analysis by Hume (1740).

2"In early tales the golem was usually a perfect servant, his only fault being a too literal or mechanical fulfillment of his master's orders [...] It was the basis for Gustav Meyrink's novel Der Golem (1915) and for a classic of German silent films (1920)." (From "Golem", in the Encyclopaedia Britannica).

3Following Hume, Sugden (1986) suggests that the punishment of defectors tends to become an ethical value, thus reinforcing its viability as an enforcement devise. Falk et al. recently provided an experimental support to this hypothesis, to our opinion conclusive. "Our findings show that the violation of fairness principle is the most important driving force of sanctions, but, in addiction, a non-negligible part of sanctions is driven by spitefulness. We find surprisingly little evidence for strategic sanctions that are imposed to create future material benefits. While non-strategic sanctions are of major importance in our experiments, strategic sanctions seem to play a negligible role" (Falk, Fehr and Fischbacher, 2001).

4We impose the rule for players to change position in the space after each round in order to ensure that the groups are different in each round. In this case, players cannot get used to a given "neighbourhood" and are interacting with the entire population. In addition, the positioning of players in the space does not guarantee that they will all be able to play in any given round: some players may find themselves isolated and not have enough neighbours to form a group. Changing position at the end of a round also means that isolated players will get a chance to play in the following round of the simulation.

5This is the standard choice mechanism in game-theoretic economic literature. Other mechanisms may be plausible too. For instance, Eshel et al. (2000) and Uno and Namatame (1999) suppose that players observe neighbours, and imitate the successful ones. This pattern looks realistic, but the realism is easily lost if the imitation is simulated too simply. To our opinion both expectation and imitation deserve consideration. We hope to include this feature in further versions of our model.

6We would like to stress that there is no pejorative meaning in assuming that immigrants do not cooperate. This feature is made necessary by three logical steps. First, what we want to study is the robustness of cooperating conventions, so we must suppose one to be in force; second, the arrival of foreigners must be modelled, in our study, as the arrival of players adopting a different convention; third, it is much simpler to suppose, in a PD setting, that this convention imposes non-ccoperation.

7Each player occupies a position in the world, and can have at most 8 neighbours (each cell is surrounded by 8 other cells, which may or may not contain a player). During each round, each player tries to be part of a group of size n. This group is composed of his neighbours, and their neighbours, etc until it reaches the size n. Hence this grouping method is called "Chain of Neighbours" in the program, as all players in a group must touch at least another player by one side or one corner of the cell. Players participate in one group only in each round. Therefore, if a group cannot reach size n, its members stay inactive for the current round (we say that they "sleep" for a round).

8Note that this behaviour occurs in our model because punishment is present and significant. A smaller or zero punishment would teach Natives to defect, or lead to a mixed equilibrium.

9We do not include the Ghetto results as they are less informative. In the case of Normal players, the Immigrants learn to cooperate through interactions with Natives on the border of the Ghetto, but cooperation only diffuses throughout the Ghetto if the number of Immigrants is small (about 100 players, 5% of Immigrants). In the large population we consider, Normal players only learn to cooperate occasionally, if they happen to meet Natives in consecutive rounds. The "more intelligent" players (Intelli, Clever and Nobel) all evolve the same strategy: they learn that if Natives are in the group (i.e. they are playing on the border of the Ghetto), it is worth for them to cooperate, otherwise, they keep on defecting.

10As the "players" (immigrant and natives) are positioned randomly on the space at the beginning of the game, and move randomly afterwards, their observed behaviour depends also on the random seed in use at the beginning of the game. Running simulations changing the seed allows to see how much this behaviour depends on the randomness part of the game. In all cases, the variance observed was very small (of the order of 1 round).

11The program is available on request from the corresponding author.


* References

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